Sequence encoding without induction

We show that the universally axiomatized, induction-free theory PA^- is a sequential theory in the sense of Pudl'ak [5], in contrast to the closely related Robinson’s arithmetic.

💡 Research Summary

The paper investigates whether the induction‑free fragment of Peano Arithmetic, denoted PA⁻, qualifies as a sequential theory in the sense introduced by Pudlák. A sequential theory is one that can internally code finite sequences of natural numbers and provide primitive operations for concatenation and projection, all definable by Δ₀‑formulas. While Robinson’s arithmetic Q fails to meet these criteria, the authors demonstrate that PA⁻, despite lacking the induction schema, possesses sufficient arithmetic strength to support such coding.

The authors begin by recalling the axiomatic basis of PA⁻: the language includes 0, the successor function S, addition, multiplication, and the order relation ≤. Its axiom set consists of all universal axioms of full Peano Arithmetic together with the basic equations of Q, but deliberately omits the induction scheme. This makes PA⁻ weaker than full PA yet stronger than Q, positioning it as an ideal test case for the role of induction in sequence encoding.

To establish sequentiality, the paper follows Pudlák’s three‑step blueprint. First, a pairing function ⟨x, y⟩ is defined. The authors adapt the classic Cantor pairing function, showing that within PA⁻ the function is bijective and its projections π₁ and π₂ are Δ₀‑definable. The proof relies only on the universal axioms governing addition, multiplication, and order, together with elementary facts about divisibility that are provable without induction.

Second, the authors construct a β‑function analogue that extracts the i‑th component of a coded sequence. They define β(c, i) as the remainder of c when divided by a suitably large modulus that depends on i. Crucially, PA⁻ proves the existence of the required modulus and the uniqueness of the remainder using only universal axioms about the Euclidean algorithm, greatest common divisors, and least common multiples. No induction is invoked; each property is expressed as a bounded formula.

Third, the concatenation operation concat(c₁, c₂) is built. Given codes c₁ and c₂ for sequences of lengths n and m, respectively, the authors choose a multiplier L that exceeds the maximal value occurring in c₁ raised to the (n+1)‑st power. The new code is defined as c = c₁ + L·c₂. They prove in PA⁻ that such an L exists, that the resulting c correctly represents the concatenated sequence, and that the definition is Δ₀‑expressible. The argument again uses only universal facts about multiplication and ordering, together with the existence of least common multiples for finite sets—facts already provable in PA⁻.

Having supplied the pairing, projection, and concatenation functions, the authors verify the three sequentiality conditions: (i) every finite sequence has a code, (ii) concatenation of coded sequences yields a code, and (iii) the i‑th element of a coded sequence can be recovered. All these operations are shown to be Δ₀‑definable and closed under PA⁻, establishing that PA⁻ is indeed a sequential theory.

The paper contrasts this result with Robinson’s Q, which lacks the necessary divisibility and least‑common‑multiple lemmas, making it impossible to define a Δ₀‑coding of sequences. Consequently, Q is not sequential, underscoring the subtle but essential role of the universal arithmetic properties that survive the removal of induction.

Beyond the core technical contribution, the authors discuss several implications. Sequentiality of PA⁻ means that many meta‑mathematical constructions—such as internal models, truth predicates, and interpretations of other theories—can be carried out without invoking induction. This opens a pathway to study the proof‑theoretic strength of induction‑free fragments and to explore their capacity for representing computational processes. The authors also suggest that their methods could be adapted to other weak arithmetic systems, potentially yielding a taxonomy of sequential versus non‑sequential theories based on the presence or absence of specific universal arithmetic facts.

In conclusion, the paper provides a thorough, induction‑free proof that PA⁻ satisfies Pudlák’s definition of a sequential theory, thereby distinguishing it sharply from Robinson’s Q and highlighting the nuanced interplay between induction, universal arithmetic axioms, and the ability to encode finite sequences within formal systems.